# Ratio between control and treatment siginificantly different than 1

I would like to know what kind of test I can use to verify my findings. In general I have a two group of people. One is a control group the other one was treated with a drug. After some period of time we collected samples from them (urine, blood, spit, etc...) and we calculated the ratio between some metabolites which can be found in those specific samples. The ratio was calculated for the metabolites grouped within the samples they were collected. For example:

Control:             Treatment:         Ratio:
Urine (Arginine)    / Urine (Arginine)   1.5
Urine (Proline)     / Urine (Proline)    1.2
Urine (Acid)        / Urine (Acid)       1.3

Blood (Tryptophane) / Blood (Tryptophane) 1.7
Blood (XX1)         / Blood (XX1)         1.3
Blood (XX2)         / Blood (XX2)         1.4


So I would like to know if a whole group (Blood, Urine, Spit) was significantly changed. I have at least 4 replicates for each of the metabolite/group. I would like to store all the ratios within replicates and perform a test which will tell me if the ratio is significantly changed to the one of the sides. What kind of a test can be used ?

• Why do you divide by the treatment results? Usually, we consider the results of the control group the "normal" ones and the treatment results in units of the normal level. (If might be intentional but nevertheless the most annoying errors in statistics are of the kind "I confused pre and post") Jan 30, 2017 at 14:05

It is only a matter on your preferred interpretation. The fact --the treatment has some effect on the metabolites-- isn't altered by the question if you prefer to interpret ratios rather than differences. The patients well-being due to the treatment doesn't depend on you interpreting ratios or differences. Yet the formulation of the hypothesis either as "no difference" or "ratio = 1" is equivalent.

So you may use a paired t-test or a Wilcoxon-test (Wilcoxon sign test). Any statistical analysis software provides these test procedures. Note that you might need some multiplicity adjustment as you have multiple metabolites.

You don't have per patient ratios anyway because each patient is either part of the control or the treatment group (if I didn't misunderstand). You can calculate and interpret the ratios of the group means, but you cannot calculate the mean of ratios.

• Let's say that I pointed it wrong. I could not share my real example because of confidence reason. I would need to specify that patient belongs to both groups (control and treatment). This case involves clones... Sorry for leading you in wrong direction. Could you write a bit more about applying the test which you specifed ? Should I create a vector of number 1 with the same length as the ratios vector to perform t-test ? Jan 30, 2017 at 16:49

In general, if you are specifically interested in the ratio of two continuous variables, I would recommend log transforming both variables and look at the difference between the log-transformed variables. As these remain continuous variables you can still use 'regular' tests for continuous variables (such as a t-test, but mind the assumptions) to see whether the difference between the transformed variables is different from 0, which would be equivalent to testing whether the ratio is different from 1 (i.e. the ratio can be found by exponentiating the difference between the transformed variables).

EDIT: After seeing @horst-grünbusch 's answer (in which he took a closer look at the data than I initially did), I'd like to note that I assumed control and treatment measurements were somehow paired. This pairing could have occurred by matching or by a pre-/post design of data collection. (The latter suggests repeated measures within individuals where the measurements were performed pre and post treatment within an individual)